An improved visualization approach in many-objective optimization

In a high-dimensional objective space, visualization of population in an approximate Pareto front is crucial to decision making process. By directly observing the performance of each solution, the trade-off between objectives, and distribution of approximate front, the decision maker can effectively decide which solution should be chosen from. Furthermore, visualization throughout the evolution process can also be exploited in designing effective many-objective evolutionary algorithms. Recently, a new visualization approach was developed by constructing a mapping from a high dimensional objective space into a two dimensional polar coordinate system, where a group of predefined direction vectors divide the whole space into a number of sub regions and each individual is associated with one weight vector. This method can be scalable to any dimensions, and simultaneously deal with a large number of individuals and multiple Pareto fronts for the purpose of visual comparison. It faithfully preserves shape, location, range, and distribution of Pareto front. However, distributions of Pareto front within each subregion and relations between different sub regions cannot be observed by this method. In this paper, in order to overcome this deficiency, we incorporate a modified multi-dimensional scaling (MDS) approach into this method. Experimental results show that the modified MDS is a suitable complementary to the existing method. Furthermore, the new design combined with existing visualization approach provides a comprehensive mean to visualize all important information in a many-objective optimization problem.